What are the sources of family resemblance? How do we quantify them numerically?
Heritability
Proportion of similarity in phenotypes that can be attributed to similarity in genotypes.
Model: Phenotype (P) = Genotype (G) + Environment (E)Variance decomposition \[var(P) = var(𝐺) + var(𝐸)\] Proportion of variance \[h^2 = \frac{var(𝐺)}{var(𝑃)}\]
Tenesa, A., Haley, C. The heritability of human disease: estimation, uses and abuses . Nat Rev Genet 14 , 139–149 (2013). doi:10.1038/nrg3377
Visscher, P., Hill, W. & Wray, N. Heritability in the genomics era — concepts and misconceptions . Nat Rev Genet 9 , 255–266 (2008). doi:10.1038/nrg2322
How to estimate heritability from data
Plot of child (offspring) height versus the average of their parents’ heights. What is a statistic that can be used to summarise the relationship between these two variables?
How to estimate heritability from data
\(\beta = \frac{\mathrm{cov}(A, B)}{\mathrm{var}(A)}\)
Estimate the beta coefficient (slope) for a simple regression from the covariance between predictor (\(A\) ) and outcome (\(B\) ) variable divided by the variance of the predictor (\(A\) ).
Simple model of genetic and environmental effects
\[
P = G + E
\]
The phenotype value \(P\) is influenced by a genetic effect \(G\) and and environmental effect \(E\) .
For simplicity assume that \(P\) is an individual’s deviation from the average phenotype in the population.
Simple model of genomics
\[
G = d + s
\]
Each individual has two copies of the genome, one inherited from each parent.
Here we label the genome from the mother \(d\) and the genome from the father \(s\) . These are the transmitted genetic effects.
Simple model of inheritance
Each parent also has a copy of the genome which they do not pass on to their child, labelled \(d^\prime\) for the mother and \(s^\prime\) for the father. These are the non-transmitted genetic effects.
Simple model of genetics, environment, and inheritance
Phenotype (\(P\) ) value is the sum of the two genetic values plus an environmental value (\(e\) ).
Mother’s phenotype: \(P_d = d + d^\prime + e_d\)
Father’s phenotype: \(P_s = s + s^\prime + e_s\)
Child’s phenotype: \(P_o = d + s + e_o\)
Regression equation
\(\beta = \frac{\mathrm{cov}(A, B)}{\mathrm{var}(A)}\)
\(A\) = average of parents’ phenotypes\(B\) = offspring phenotype
Therefore, \(\beta = \frac{\mathrm{cov}(\frac{P_d + P_s}{2}, P_o)}{\mathrm{var}(\frac{P_d + P_s}{2})}\)
Parent–offspring covariance
\[
\mathrm{cov}(\frac{P_d + P_s}{2}, P_o)
\]
\[
= \mathrm{cov}(\frac{d + d^\prime + e_d + s + s^\prime + e_s}{2}, d + s + e_o)
\]
Parent-offspring covariance
Expand the terms. Recall that:
\[
\mathrm{cov}(A+X,B+Y) = \\
\mathrm{cov}(A,B) + \mathrm{cov}(A,Y) + \mathrm{cov}(X,B) + \mathrm{cov}(X,Y)
\] Thus we can do a pairwise expansion to: \[
= \mathrm{cov}(\frac{d}{2} + \frac{d^\prime}{2} + \frac{e_d}{2} + \frac{s}{2} + \frac{s^\prime}{2} + \frac{e_s}{2}, d + s + e_o)
\] \[
= \mathrm{cov}(\frac{d}{2}, d) + \mathrm{cov}(\frac{d^\prime}{2}, d) + \dotsm+ \mathrm{cov}(\frac{e_s}{2}, e_o)\]
Simplifications and assumptions
Covariance between a genetic effect and itself \[
\mathrm{cov}(\frac{d}{2}, d), \mathrm{cov}(\frac{s}{2}, s)
\]
Covariance between genetic effects from the same parent \[
\mathrm{cov}(\frac{d^\prime}{2}, d), \mathrm{cov}(\frac{s^\prime}{2}, s)
\]
Covariance between genetic effects from different parents \[
\mathrm{cov}(\frac{d^\prime}{2}, s), \mathrm{cov}(\frac{s^\prime}{2}, d)
\]
Simplifications and assumptions
Covariance between parent and offspring environment effects \[
\mathrm{cov}(\frac{e_d}{2}, e_o), \mathrm{cov}(\frac{e_s}{2}, e_o)
\]
Covariance between parental genetic and offspring environmental \[
\mathrm{cov}(\frac{d}{2}, e_o), \mathrm{cov}(\frac{s}{2}, e_o)
\]
